Testing nonlinear stochastic models on phytoplankton biomass time series

Time series of algae mass in German water reservoir lakes being collected for several years in irregular sampling, half weekly on average, are tested in their stationary distribution against several univariate stochastic processes with linear and nonlinear dynamics, with pure additive and with state dependent noise. In each case the parameters of the corresponding Fokker–Planck equations in the stationary state are estimated by maximisation of the log-likelihood. For statistical evaluation the Kolmogorov–Smirnov-test is applied with additional graphical tools, being especially suited for short data sets typical for ecological systems. To obtain the optimally adapted stochastic process, an additional parameter has to be estimated describing the autocorrelation. We use two methods: first, we use a novel method based on the estimation of conditional probabilities directly from the data. The resulting stochastic model with state dependent noise is simulated. Second, we use the Lomb normalised Fourier spectrum for unevenly sampled data, e.g. no measurements during winter time to minimise the overall deviation between the empirical spectrum and spectra obtained from stochastic simulations, the sampling of which follows in detail the one of the empirical data. The parameter value of the first method is used as a starting point for the adaptation of the cumulative spectrum. In this way a stochastic process is obtained which describes the stationary distribution as well as the spectrum of the empirical data both indistinguishable at high statistical confidence.